Scientific Notation: Writing Very Large and Very Small Numbers

The form

$$a \times 10^n \qquad \text{where } 1 \le a < 10$$

Two parts: a number between $1$ and $10$, and a power of ten that says how big it really is.

$$4{,}500{,}000 = 4.5 \times 10^6$$

The $4.5$ carries the digits that matter. The $10^6$ carries the size.

Standard form	Scientific notation
$300$	$3 \times 10^2$
$52{,}000$	$5.2 \times 10^4$
$0.007$	$7 \times 10^{-3}$
$0.00061$	$6.1 \times 10^{-4}$

Why it is worth using

Compare these two ways of writing the same three facts.

Quantity	Standard form	Scientific notation
Distance to the Sun	$149{,}600{,}000$ m	$1.496 \times 10^{11}$ m
Width of a human hair	$0.00007$ m	$7 \times 10^{-5}$ m
Mass of an electron	$0.000000000000000000000000000000911$ kg	$9.11 \times 10^{-31}$ kg

The last row is the argument on its own. Counting thirty zeros correctly is a task nobody performs reliably, and one miscounted zero changes the answer by a factor of ten.

Scientific notation also makes comparison immediate. Which is bigger, $3 \times 10^8$ or $9 \times 10^7$? Compare the exponents first: $10^8$ beats $10^7$, so the first is larger — about three times larger — and that took one glance rather than counting digits in $300{,}000{,}000$ against $90{,}000{,}000$.

This is why the notation is standard in science. A quantity’s order of magnitude — its power of ten — is often the part that matters, and this form puts it in plain view instead of hiding it in a run of zeros.

Converting to scientific notation

Move the decimal point until exactly one nonzero digit sits in front of it, then count the moves.

$$4{,}500{,}000 \;\rightarrow\; 4.500000 \qquad \text{six places left}$$

Moving left made the front number smaller, so the power of ten must make it bigger again:

$$4.5 \times 10^6$$

For a small number the point moves the other way:

$$0.00032 \;\rightarrow\; 3.2 \qquad \text{four places right}$$ $$3.2 \times 10^{-4}$$

The exponent’s sign records the direction. Left is positive, right is negative — and the check is always the same: does the answer describe a number of the right size?

Converting back to standard form

Reverse it. A positive exponent moves the point right, a negative one moves it left, padding with zeros:

$$3.2 \times 10^5 = 320{,}000$$ $$6.1 \times 10^{-4} = 0.00061$$

Counting the zeros in the second: the point moves four places left from $6.1$, so three zeros sit between the decimal point and the $6$.

Multiplying and dividing

This is where the notation earns its place, because the exponent rules apply directly.

$$(3 \times 10^4)(2 \times 10^5) = (3 \times 2) \times 10^{4+5} = 6 \times 10^9$$

Front numbers multiply, exponents add. Dividing subtracts instead:

$$\frac{8 \times 10^7}{4 \times 10^3} = 2 \times 10^{4}$$

Sometimes the front number lands outside $1$ to $10$ and needs one more step:

$$(5 \times 10^3)(4 \times 10^6) = 20 \times 10^9$$

$20$ is too big, so rewrite it as $2 \times 10^1$ and combine the powers:

$$2 \times 10^1 \times 10^9 = 2 \times 10^{10}$$

Worked examples

Common mistakes

Practice problems